"What If..." and Discrete Variables To Help Solve Nonlinear Programming Problems: Another Illustration

Second Edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following mathematical formulation in Wang, Zhang, and Gao [99, p. 1514, Example 3], which is as follows:

Minimize

X(1)

subject to

.274 * X(4) * X(5) ^ 4 + 2520.66 * X(2) * X(5) ^ 5 + X(1) * X(4) ^ 2 - X(1) * X(2) * X(3) * X(4) + 1 <= 1

X(2) * X(3) ^ -1 * X(4) <= 1

X(2) * X(5) ^ 4 <= 1

X(4) * X(5) ^ 3 <= 1

10 ^ -12 <= X(1) <= 2

20<= X(2) <= 35

120 <= X(3) <= 160

1 <= X(4) <= 10

10 ^ 6 <= X(5) <= 1.

0 DEFDBL A-Z

1 REM DEFINT I, J

2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

79 FOR JJJJ = -32000 TO 32000 STEP .01

89 RANDOMIZE JJJJ

90 M = -3D+30

95 A(1) = (100 * (1D-12 + RND * (2 - 1D-12))) / 100

97 A(2) = INT(100 * (20 + RND * 15)) / 100

98 A(3) = INT(100 * (120 + RND * 40)) / 100

99 A(4) = INT(100 * (1 + RND * 9)) / 100

100 A(5) = (100 * (1D-06 + RND * (1 - 1D-06))) / 100

124 FOR I = 1 TO 4000

129 FOR KKQQ = 1 TO 5

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 5)

144 IF RND < .5 THEN 160 ELSE GOTO 166

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + FIX((RND ^ (RND * 15)) * r)

165 GOTO 168

166 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

168 NEXT IPP

191 IF X(1) < 1D-12 THEN 1670

192 IF X(1) > 2 THEN 1670

193 IF X(2) < 20 THEN 1670

194 IF X(2) > 35 THEN 1670

195 IF X(3) < 120 THEN 1670

196 IF X(3) > 160 THEN 1670

197 IF X(4) < 1 THEN 1670

198 IF X(4) > 10 THEN 1670

199 IF X(5) < 1D-06 THEN 1670

200 IF X(5) > 1 THEN 1670

205 IF X(2) * X(5) ^ 4 > 1 THEN 1670

206 IF X(4) * X(5) ^ 3 > 1 THEN 1670

208 IF X(2) * X(3) ^ -1 * X(4) > 1 THEN 1670

277 IF .274 * X(4) * X(5) ^ 4 + 2520.66 * X(2) * X(5) ^ 5 + X(1) * X(4) ^ 2 - X(1) * X(2) * X(3) * X(4) + 1 > 1 THEN 1670

463 PD1 = -X(1)

466 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 5

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT I

1891 IF M < -1D-05 THEN 1999

1899 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. The complete output of one run through JJJJ=

-26174.04000093257 is shown below:

3.457070396970928D-06 26.81 121.14 3.87

3.604132965558767D-02 -3.457070396970928D-06 -31841.3100000254

**1.311302685103044D-06 22.19 131 5.220000000000001**

** 2.179428408056498D-02 -1.311302685103044D-06 -31183.17000013075 **

3.337860607466325D-06 34.47 136.08 3.03

7.472077123641968D-03 -3.337860607466325D-06 -26174.04000093257

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM, 64-bit Operating System, and QB64v1000-win [102], the wall-clock time (**not** **CPU time**) for obtaining the output through

JJJJ = -26174.040000993257 was 45 minutes, counting from "Starting program...". One can compare the computational results above with those in Wang, Zhang, and Gao [99, p. 1515, Table 1, Example 3].

**Acknowledgement**

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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